Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 1G / b / Solution

The discriminant isThe discriminant criterion for a quadratic congruence modulo an odd prime therefore says that, for $p\ne 3$, a solution exists exactly whenThe quadratic character of minus three giveswhich equals $1$ precisely for $p\equiv 1(\mathrm{mod}3)$. When $p=3$, the polynomial isso it also has a solution. Hence